...........

I escaped first quarter with a 90.

I believe no one else in the history of my class has averaged a 90 on the first quarter.

They should make a happy birthday emoticon.

(28,45,53) + 3(8,15,17) = (51,52,53)

I believe that no one from Massachusetts is still taking the contest with the above problem, so it is safe to be released to the public.

Obviously, no one has memorized (28,45,53) so what more can I say?

In the spirit of this blog and the blog entry a few back, these are actually easy problems.

These are from Problems From the Book
paraphrased unless said otherwise
i give credit to Problems from the book for compiling these problems and in some cases for writing the problems.

These are all digit sum questions, and is this function.

Find a 100-digit number divisible by the sum of its digits.

Are there arbitrarily long arithmeric sequences whose terms have the same digit sum? What about infinite arithmetic sequences (taken exactly from the book)

If and , find (taken exactly from the book)

Seems easy, but i'll have to look at it: Are there 19 positive integers with the same digit sum, which add up to 1999? (taked exactly from the book)

computation heavy, i bet
Prove that
where N can be arbitrarily large and .
Is it true that the sum is also less than 3?

Find 50 distinct natural numbers s.t. the sum of each natural number and its digit sum is invariant for the 50 numbers (like 99 and 108 for 2 numbers)
actually, just prove that it's possible. the explicit numbers aren't very easy to express probably.

Prove that one of is a multiple of 11 for

Does there exist a integer whose digit sum is 1000 and whose square has a digit sum of 1000000?

Prove that there are infinitely many natural numbers such that

(poorly paraphrased)




Tricky, haven't decided if it's easy or not
Are there arbitrarily long arithmetic sequences of non-Niven numbers; are there infinitely long arithmetic sequences of non-Niven numbers?

ok a strong theorem kills the first part, but try to solve it a different way. i'm not sure if there is an easy way yet, though.

Find the next 100 numbers

8, 6, 7, 9, 5, 6, 8, 6, 8, 4, 6, 9, 8, 7, 10, 6, 8, 5
hopefully that's right.

EDITs: searching 8, 6, 7 atm shows 666 results (see next edit)
hooray, this is not in the oeis!

1,2,3,2,1,2,3,4,2,1,2,3,4,3,2,3,4,5,3,2

Find the next term.

Should not be that hard with the technology currently available (hint hint JBL).